Existence Of Duadic Codes Over Z₄

Duadic codes form an important class of linear codes for both theoretical and practical reasons in error-correcting codes. They were first introduced by Leon et. al(1984) as generalized quadratic residue cyclic codes over fields. Rushanan(1991) generalized them to duadic abelian codes and Zhu(1996) further generalized them to duadic group algebra codes. Many properties of quadratic residue codes can be perserved by these generalizations, for example, the square root bound and connection with projective planes. Zhang(1997) further generalized them to central duadic group algebra codes. About the existence of duadic codes, Ward and Zhu(1994) obtained necessary and sufficient conditions for existence of duadic abelian codes, Zhang(1997) gave the existent conditions for such codes by some parameters and obtained necessary and sufficient conditions for existence of central duadic group algebra codes. Recently, duadic codes over Z₄ are presented by Langevin and Sole(2000), but the existent problem of duadic abelian codes over Z₄ hasn't been solved yet.This thesis solves the existent problem of duadic abelian codes over Z₄. A survey of duadic codes is presented firstly, necessary and sufficient conditions for the existence of nontrivial duadic abelian codes over Z₄ are presented secondly. Finally, necessary and sufficient conditions for the existence of nontrivial duadic abelian codes over Z₄ with splitting μ-1 are presented, these conditions are also necessary and sufficient for the existence of nontrivial self-dual abelian codes over Z₄.